1. Field of the Invention
The present invention relates to monitoring and analysis of both balanced and unbalanced three-phase electrical power transmission and distribution networks, and more particularly to a system and method for determining the grid state of such a network by determining the power flows using a deterministic, non-iterative, real time analysis of the network.
2. Description of the Background
The main objective of a three-phase power flow analysis is to determine with sufficient accuracy the voltages and flows in the three phases of the network, in order to calculate the degree of imbalance present in the system. Although most three-phase power systems are designed and built in a balanced configuration, there are certain types of loads that produce unbalanced conditions. Important examples are high-speed trains and AC arc furnaces. Imbalances in the network cause overheating in rotating machines, and displacements of zero-crossings in static power converters, generating harmonics that in turn cause other types of problems to equipment connected to the network. Imbalances are one of the main criteria in evaluating power quality supply, and utilities strive to minimize it. Therefore, the three-phase power flow analysis is the main tool for the study and monitoring of imbalances. It is also the first step of a more complex analysis known as three-phase harmonic power flow, in which the network is analyzed for the presence and propagation of voltages and currents with frequencies that are higher-order harmonics of the fundamental one.
The main subject of three-phase power flow studies are distribution networks, where the degree of imbalance is potentially larger. Transmission networks are always operated with negligible imbalances, unless there is an unbalanced fault condition. Consequently, many three-phase power flow methods have focused on distribution networks and have tried to take advantage of the mostly-radial (i.e. tree-like) structure they have. However, the current trend towards the so-called smart-grid, with smaller, distributed generation such as solar, wind and other renewables, is changing distribution networks to be more meshed in nature, with more branches in which the direction of flow may reverse. Therefore, the power flow methods used for distribution networks may no longer make assumptions about the radial structure, and have to use the general-purpose calculations for meshed networks, as in transmission grids.
Early three-phase load flow methods were rather straightforward extensions of the conventional Newton-Raphson methods used for one-phase circuits; see for example R. G. Wasley et al. 1974, or J. Arrillaga et al. 1978. X. P. Zhang 1996 describes methods to calculate the load flow equations in terms of symmetrical components, dubbed Sequence Decoupling-Compensation Newton-Raphson and Sequence Decoupling-Compensation Fast Decoupled, where the underlying technique for the solvers is either Newton-Raphson or Fast-Decoupled Newton-Raphson, respectively. The hybrid method described by B. K. Chen et al. 1990, based on Gauss-Seidel iteration, combines an implicit Z-bus treatment with the more traditional bus admittance methods, in order to improve convergence on very long radial systems typical of distribution networks. U.S. Pat. No. 5,734,586 to Chiang et al. shows a method to control distribution networks with radial structure and imbalances, in order to achieve steady state in real-time.
C. S. Cheng et al. 1995 extend to three-phase systems the so-called compensation method, which exploits the near-radial (i.e. weakly meshed) nature of distribution networks by means of an iterative technique that sweeps backward and forward across the tree topology. A method that allows the combination of phase and symmetrical components, based on PQ residuals, is shown in B. C. Smith et al. 1998. An alternative formulation based on current residuals is proposed in P. A. N. Garcia et al. 2000.
Other methods have been proposed in order to improve the flexibility in the modeling and wiring configuration of PQ buses, such as M. Valcarcel et al. 1993; and U.S. Pat. No. 7,209,839 to Roytelman, which shows a way to deal specifically with loads on “floating” delta-connected transformers. Although not strictly related to solving a power flow problem, U.S. Pat. No. 7,584,066 to Roytelman discloses a method to estimate individual feeder loads when the only information available is the total substation PQ injection and the individual feeder current magnitudes.
The method of W. Xu et al. 1991 offers greater flexibility in the modeling of both PQ and PV buses. A method combining the formulation based on current residuals, flexible treatment of PQ/PV buses, and symmetrical components, is presented in W. Xu et al. 2009.
It is pertinent to point out that all the methods cited above are iterative in nature. Even the most recent advances in power flow-solving techniques for one-phase networks (which eventually are extended to three-phase networks) involve iterative schemes. See for instance U.S. Pat. No. 7,769,497 to Patel, which discloses a method to improve the convergence rate of the iteration; and U.S. Pat. No. 7,813,884 to Chu et al., which discloses a method to include modern FACT devices such as STATCOMs and UPFCs in a unified way while preserving the quadratic convergence rate of Newton-Raphson.
The shortcoming of all previous-art methods is that, since the mathematical equations of a power flow problem are multi-valued, iterative techniques cannot guarantee that the iteration does not converge to a spurious solution (one in which one or more buses end up on the lower branch of their nose-like P-V curve). Methods that employ numerical continuation (see for instance Tolikas et al. 1992, C. A. Canizares et al. 1993, and Guo et al. 1993; see also U.S. Pat. No. 5,745,368 to Ejebe et al.) somehow alleviate this problem, but they cannot eliminate it completely because of the fractal nature of the attractor sets. Moreover, such methods are computationally costly and they are mostly oriented to the precise calculation of voltage collapse points, not for general power flow computation.